Extrapolation and local acceleration of an iterative process for common fixed point problems

We consider sequential iterative processes for the common fixed point problem of families of cutter operators on a Hilbert space. These are operators that have the property that, for any point x∈H, the hyperplane through Tx whose normal is x−Tx always “cuts” the space into two half-spaces, one of which contains the point x while the other contains the (assumed nonempty) fixed point set of T. We define and study generalized relaxations and extrapolation of cutter operators, and construct extrapolated cyclic cutter operators. In this framework we investigate the Dos Santos local acceleration method in a unified manner and adopt it to a composition of cutters. For these, we conduct a convergence analysis of successive iteration algorithms.